The space far from a black hole is kind of boring. It
has no distinguishing features besides the degree to which it is bent, and
this bending, is no different than that of an ordinary star of the same
mass. In fact there is a "no hair" theorem that guarantees black holes to be virtually
featureless when viewed from far away. All the bumps and wriggles of the matter from
which they were formed are smoothed out as the matter contracts, so that the final shape of the horizon is always perfectly smooth and round.
Near the event horizon, things are more interesting. To a distant observer,
events near the horizon appear to slow down. If you drop a clock into a black hole
it appears to tick more and more slowly as it approaches the event horizon.
Time actually appears to stop right at the horizon.
The clock's motion towards the black hole also slows down and to a distant
observer it takes literally forever to fall through.
If you are unfortunate enough to be falling with the clock, time appears to progress
normally. You fall through the horizon in a relatively short time, and once you are
past it, you get sucked to the singularity at the center in a millionth of a second
(for a solar mass black hole). Time and space interchange roles, and you can no more avoid falling to the center than you can avoid moving from the present into the future.
The only 100% reliable way to detect
the presence of a black hole is to fall through the horizon and verify that
it is literally impossible to stop moving towards the center. Of course your discovery
won't do your career much good: there would be no way to publish your results.